Are these two versions of back-propagation equivalent?
The two examples present essentially the same operation:


In both cases, the network is trained with gradient descent using the backpropagated squared error computed at the output.
Both examples use the logistic function for node activation (the derivative of the logistic function $s$ is $s(1 - s)$. This derivative is obviously very easy to compute, and this is part of the reason why it was so widely used (these days the ReLU activation function is more popular, especially with convolutional networks).
The first method also uses momentum.


The main difference I can see is that in the first case backpropagation is iterative while in the second example it is performed in batch mode.

The last video in the series by Welch Labs introduces a quasi-Newtonian method which offers the advantage of finding the minimum of the cost function by computing the Hessian (matrix of second-order derivatives of the error with respect to the weights). However, this feels like comparing apples and oranges - the vanilla gradient descent does not use second-order information.